Erdös–Hajnal conjecture for new infinite families of tournaments

Abstract

AbstractErdös–Hajnal conjecture states that for every undirected graph there exists such that every undirected graph on vertices that does not contain as an induced subgraph contains a clique or a stable set of size at least . This conjecture has a directed equivalent version stating that for every tournament there exists such that every ‐free ‐vertex tournament contains a transitive subtournament of order at least . This conjecture is known to hold for a few infinite families of tournaments. In this article we construct two new infinite families of tournaments—the family of so‐called galaxies with spiders and the family of so‐called asterisms, and we prove the correctness of the conjecture for these two families.